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Ordered pair

An ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element. An ordered pair with first element a and second element b is usually written as (a, b).

(The notation (a, b) is also used to denote an open interval on the real number line; context should make it clear which meaning is meant. To distinguish the two meanings, the ordered pair (a,b) is sometimes written as $\langle a,b\rangle$.)

Two ordered pairs (a1, b1) and (a2, b2) are equal if and only if a1 = a2 and b1 = b2.

The set of all ordered pairs whose first element is in some set X and second element in some set Y is called the Cartesian product of X and Y, and written X × Y. Subsets of X × Y are binary relations.

Ordered triples and n-tuples (ordered lists of n terms) are defined recursively from this definition: an ordered triple (a,b,c) can be defined as (a, (b,c) ): two nested pairs . This approach is mirrored in programming languages: It is possible to represent a list of elements as a construction of nested ordered pairs. For example, the list (1 2 3 4 5) becomes (1, (2, (3, (4, (5, {}))))). The Lisp programming language uses such lists as its primary data structure.

In axiomatic set theory, where all mathematical objects are given set-theoretic definitions, the ordered pair (a, b) is usually defined as the Kuratowski pair (a, b)K := {{a}, {a,b}}. The statement that x is the first element of an ordered pair p can then be formulated as

Yp : xY

and that x is the second element of p as

(∃ Yp : xY) ∧ (∀ Y1p, ∀ Y2p : Y1Y2 → (xY1xY2)).

Note that this definition is still valid for the ordered pair p = (x,x) = { {x}, {x,x} } = { {x}, {x} } = { {x} }; in this case the statement (∀ Y1p, ∀ Y2p : Y1Y2 → (xY1xY2)) is trivially true, since it is never the case that Y1Y2.

Notice that the set {1, 2} is the ordered pair (0, 1), since 1={0} and 2={0, 1}.

The above definition of an ordered pair is "adequate", in the sense that it satisfies the characteristic property that an ordered pair must have (namely: if (a,b)=(x,y), then a=x and b=y), but also arbitrary, as there are many other definitions which are no more complicated and would also be adequate. Examples for other possible definitions include

1. (a,b)reverse:= { {b}, {a,b} }
2. (a,b)short:= { a, {a,b} }
3. (a, b)01:= { {0,a}, {1,b}}

The "reverse" pair is almost never used, as it has no obvious advantages (nor disadvantages) over the usual Kuratowski pair. The "short" pair has the disadvantage that the proof of the characteristic pair property (see above) is more complicated than for the Kuratowski pair (the axiom of regularity has to be used); moreover, as the number 2 is in set theory sometimes defined as the set { 0, 1 } = { {}, {0} }, this would mean that 2 is a pair, 2 = (0,0)short. Definition 3 is considered bad taste because why should 0 and 1 play special roles?

03-10-2013 05:06:04